Question

The annual Gross Domestic Product (GDP) of a country is the value of all of the goods and services produced in the country during a year. During the period 1985-1999, the Gross Domestic Product of the United States grew about 3.2% per year, measured in 1996 dollars. In 1985, the GDP was $577 billion. I what year did/or will the GDP equal $1.6 trillion?

Answer 1

The increase indicates a compound growth of 3.2% per year.
The multiplier is given by 100%+3.2% = 103.2% = 1.032

The initial amount of GDP is 577 billion

We are looking to find in how many years since 1985 that the GDP equals to 1.6 trillion, this is  1.6×10³ billion

The formula for a compound growth/decay is given as
$A_{n}= A_{o} (Multiplier)^{n}$, where $A_{n}$ is the final value and  $A_{o}$ is the intial value.

We are looking to find 'n'

1.6×10³ = 577(1.032)ⁿ
(1.6×10³) ÷ 577 = 1.032ⁿ
1060/577 = 1.032ⁿ ⇒ take log both sides
log( 1060/577 ) = log (1.032)ⁿ
log (1060/577) ÷ log (1.032) = n
n = 19.31 ≈ 19 years

The year when GDP achieves 1.6 trillion is 19 years from 1985, which will be in 2004, providing the increase rates stays the same.